使用Arima（）预测平稳序列

Question

I am working on a forecasting exercise. The preferred model is an ARIMA (0,0,1) (0,1,1)4 in which there are three exogenous variables (Forestalling.1, Forestalling.2, Break).

My dependent variable is Pmean , the average house price, and the exogenous variables are dummy variable that indicate changes in the legislation and the property crisis (these variables are made of the following values 0, 1, -1).

My initial approach was to differentiate the original and fit the Arima model; however this causes me trouble when trying to forecast the series as the forecast is done on the stationary series - diff(log(x$Pmean),4)

fit = Arima(diff(log(x$Pmean),4),
      order=c(1,0,0),
      seasonal=list(order=c(0,0,1), period =4),
      xreg=xregvariables)

               Estimate Std. Error z value  Pr(>|z|)    
ar1             0.686212   0.128593  5.3363 9.485e-08 ***
sma1           -0.583000   0.110908 -5.2566 1.467e-07 ***
intercept       0.101515   0.010318  9.8386 < 2.2e-16 ***
Forestalling 1  0.035008   0.011365  3.0804  0.002067 ** 
Forestalling 2 -0.033731   0.013151 -2.5649  0.010320 *  
Break          -0.087386   0.013113 -6.6640 2.664e-11 ***

AIC=-216.75

I tried to fit an alternative model in which I incorporate the seasonal differences, but the results are not optimal, and my estimates are not significant. They even return a different directions for some of the parameters (Forestalling2); it has a negative effect in the original model and a positive (null) effect in the second.

  fit = Arima(log(x$Pmean)
       order=c(1,0,0),
       seasonal=list(order=c(0,1,1), period =4),
       xreg=xregvariables ,
       include.drift = TRUE)

z test of coefficients:

                  Estimate  Std. Error z value  Pr(>|z|)    
ar1             0.97042096  0.03430919 28.2846 < 2.2e-16 ***
sma1           -0.53044592  0.13689248 -3.8749 0.0001067 ***
drift           0.01407096  0.01016345  1.3845 0.1662158    
Forestalling 1  0.03475176  0.01210626  2.8706 0.0040974 ** 
Forestalling 2  0.00094803  0.01343471  0.0706 0.9437434    
Break          -0.01077423  0.02376049 -0.4535 0.6502236   

AIC=-206.84

Would anyone know if it is possible to retransform the series in my first model so I can forecast the original series log(x$Pmean) or x$Pmean using the model estimates?

If that it not possible, is it possible to internalize the differencing in the second Arima model and have the same results model as the first model?

Thanks

Answer 1

In the first first Arima model,

fit=Arima(diff(log(x$Pmean),4),
  order=c(1,0,0),
  seasonal=list(order=c(0,0,1), period =4),
  xreg=xregvariables)

it looks like you are taking the first seasonal difference manually. This should be equivalent to the Arima model,

fit=Arima(log(x$Pmean),
  order=c(1,0,0),
  seasonal=list(order=c(0,1,1), period =4),
  xreg=xregvariables)

which takes the seasonal first difference. Then you can use the "fpp" package to automatically form forecasts for the logged data.

library(fpp)
forecast(fit,h=12,xreg=x_test)